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Phys 498A Lecture NotesThursday, January 30, 1997
Lecturer: Erik Koch
If we let k2(x) = (2m/
h) [ E - V(x) ],
the Schrödinger equation takes the form
We want to discretize this to a grid with spacing h. The second derivative operator can be discretized as
where we have used the notation fj = f(xj) with the xj's being the grid points.
Direct application of the discritized derivative leads to a discretized Schrödinger equation with errors of order O( h4).
This could be solved for uj+1 and used to integrate the equation. However, with a little extra work we can get a method that has errors of order O( h6 ), a substantial improvement known as the Numerov Method.
The discretized 2nd derivative formula is
Thus the Schrödinger equation becomes